“linearity” in arithmetic differential equationsmath.huji.ac.il/~dupuy/notes/ucsd.pdf ·...
TRANSCRIPT
“Linearity” in Arithmetic Differential
Equations Taylor Dupuy
UCSD Number Theory Seminar
Part 1: Origins/History
Part 2: Arithmetic Differential Equations
Part 3: “Linear” Diff ’l Eqns
PART 1
Origins of Arithmetic Differential Algebra
A/K
Sni=1(Ai + ai)
Ai = Ai(K)
Diophantine Geometryabelian variety
Ai A
ai 2 Ai(K)
configuration = finite union of translates of ab subvar
=
configurations form a basis of closed sets for a topology on called the configuration topologyA(K)
S ⇢ A(K)
S f.g. =) SConf = SZar
Mordell-Lang Problem
rf.r.
S f.g. if contained in f.g. subgroupDefn. f.r. f.r.
C/Q
g(C) � 2
Jac(C) = A
C ,! Jac(C)
C(Q) ⇢ Jac(C)(Q)
C(Q)Conf = C(Q)Zar ⇢ C(K)
#C(Q) < 1
S f.g. =) SConf = SZar
Mordell-Lang Problem
Example: (Mordell Problem)
rf.r.
C/K
K
g(C) � 2
A = Jac(C).
S = C(K) \A(K)tors
⇢ A(K)tors
SZar ⇢ C(K).
S f.g. =) SConf = SZar
Mordell-Lang Problem
example. (Manin-Mumford Problem)
rf.r.
History of Mordell-Lang Problem
• Manin - Mordell and Manin-Mumford when K = function field.
• Raynaud - Manin-Mumford for number fields
• Faltings, Coleman, Parshin - Lang Conjecture in General
• Buium - Lang Conjecture over function fields (using differential algebra), char 0
• Hrushovski - Relative Lang, char p
Finiteness hi(X) < 1
Vanishing hi(X) = 0 for i > 2n or i < 0
Trace Map Tr : H2n ⇠= K
Poincare Duality Hi ⇠= H2n�i⇤ via cup product
Kunneth H•(X ⇥ Y ) = H•(X)⌦H•(Y )
Lefschetz Axioms
Cycle Map Zi(X) ! H2i has some nice properties
Weak Lefschetz Re: pullback of hyperplane sections
Hard Lefschetz Hi ! Hi+2, ⇠ 7! ⇠ ^ !where ! = [H], H a hyperplane section.
H•: { Smooth Varieties/ k } ! { Graded K-algebras}
H•(X) =
2 dim(X)M
w=0
Hw(X)
Weil Cohomology Theory
�⌦F1 Z : SchF1 ! SchZ.
Spec(Z)F1 in the category SchF1 to get Spec(Z)F1
H•D : SchF1 ! Graded R-algebras
Setup for Desired Category
Deninger Cohomlogy Theory
Frechetness The spaces Hw(XD, j⇤C)
Vanishing Hw(X) = 0 for w > 2n or w < 0
Trace Map Tr : H2n ⇠= K
Poincare Duality Hi ⇠= H2n�i⇤ via cup product
Kunneth H•(X ⇥ Y ) = H•(X)⌦H•(Y )
Lefschetz Axioms ???, I’m not sure exactly how these should look but there should be something here.
Hodge ⇤ ⇤ : Hw ⇠= H2n�w
Real Action � : R⇥XD ! XD, which should be thought of as a replacement of the Frobenius.
H•D : SchF1 ! Graded R-algebras
Deninger Cohomlogy Theory
b⇣(s) =
Q̀b⇣(⇢)=0
s�⇢2⇡
s⇡ · s�1
⇡
.
=det1( s�✓
2⇡ |H1)
det1( s�✓2⇡ |H0)det1( s�✓
2⇡ |H2).
• ✓ acts as zero on H0
• ✓ acts as identity on H2
✓ = limt!0�t⇤�I
t
�⇤t
Deninger Says:
(induced action on cohomology)
thm
guess
hf, gi := Tr(f1 ^ ⇤f2)
f1, f2 2 H1
f1 ^ ⇤f2 = ✓(f1 ^ ⇤f2)= ✓(f1) ^ ⇤f2 + f1 ^ ✓(⇤f2)= ✓(f1) ^ ⇤f2 + f1 ^ ⇤✓(f2)
f1 [ ⇤f2 2 H2
✓ acts trivially on H2.
✓(f ^ g) = ✓(f) ^ g + f ^ ✓(f).
Pairing:
How derivatives act:
Get antisymmetry property:
Conditional Proof
bhf1, f2i = h✓(f1), f2i+ hf1, ✓(f2)i=) ✓ � 1/2 is antisymmetric on H1
=) �(✓ � 1/2|H1) ⇢ iR
=) �(✓|H1) ⇢ 1/2 + iR
Remarks
• Consistent with random matrix theory statistics
• Algebra with involution (H1(Spec(Z)F1), ⇤)
✓⇤ Noncommutative Geometric Object?
Prototype = BC system ?
Topological Hochschild Cohomology De Rham Witt
Cyclic Cohomology (NCG)
Borger-Buium Descent
X/K smooth projective
K = K char(K) = 0
� : K ! K
K� = {r 2 K : �(r) = 0}
Descent to the constants
1. KS(�) = 0
2. J1(X)
⇠=
TX as schemes over X
3. 9X 0/K�such that X 0 ⌦K� K ⇠
=
X
T.F.A.E.Theorem
CLASSICAL
Arithmetic Descent
1. DIn(X/R) = 0
2. J1(X)n
⇠=
FTXn as torsors
3. Xn/Rn admits a lift of the Frobenius.
R = W (Fp)
(FTXn)n�0
Theorem (D.)
“Canonically lifted” Frobenius Tangent Bundle
X/R
=)J1
(X)n torsor under FTXn
(*)
F1
�⌦F1 Z
Monad Geometries
Set Monoid Algebraic Geometries
Lambda Schemes
Blueprints
Generalized Rings
Sketch of Land
Connes-ConsaniMarcolliSouleBerkovichDeitmar Lorsch.
BorgerBuium
Schemes
subcat
Toen-VasquiesQuillen
Durov
�⌦F1 Z
(???)
PART 2
Wifferential Algebra
What is a p-derivation?
n� np = p · CRAP
CRAP = n�np
p
�p(n) =n�np
p
n ⌘ npmod p
8n 2 Z, 8p primeFermat’s Little Theorem
This is a p-derivation
�p(n) =n�np
p
Product Rule
Sum Rule
�p(ab) = �p(a)bp + ap�p(b) + p�p(a)�p(b)
�p(a+ b) = �p(a) + �p(b)�p�1X
j=1
1
p
✓p
j
◆ap�jbj
Zero mod p
Zero mod p
non-linear�(1) = 0
Kills Unit
p-derivations ring homomorphismsf : A ! W1(A)�p : A ! A
“dual numbers”
“Witt vectors”
“Wittferentiation”
derivations ring homomorphismsf : A ! A["]/h"2i� : A ! A
“infinitesimals”
“wittfinitesimals”
Witt Vectors.
(x0, x1)(y0, y1) = (x0y0, x1yp0 + y1x
p0 + px1y1)
(x0, x1) + (y0, y1) = (x0 + y0, x1 + y1 + Cp(x0, y0))
Cp(X,Y ) = Xp+Y p�(X+Y )p
p 2 Z[X,Y ]
�(a) ⌘ ap mod p
� : A ! B
lifts of the Frobenius ⇡ p-derivations
�(a) := ap + p�p(a)
�p : A ! B + rules
�p(a) :=�(a)�ap
p
p-torsion free
�p(ab) = �p(a)bp + ap�p(b) + p�p(a)�p(b)
�p(a+ b) = �p(a) + �p(b)�p�1X
j=1
1
p
✓p
j
◆ap�jbj
�p : A ! B
Always an -algebraA
8a, b 2 A
Defn. (Buium, Joyal 90’s)A p-derivation is a map of sets such that
R = Zp
�
p
(x) = x�x
p
p
R = Zp[⇣]
⇣
�
p
(x) = �(x)�x
p
p
�(x) =
=
(⇣ 7! ⇣p,
identity,
unique lift of the frobenius
= root of unity coprime to p
example.
example.
on roots of unity else
R = Zurp
= Zp[⇣ : ⇣n = 1, p - 1]example
�
p
(x) = �(x)�x
p
p
EXAMPLES
�p(pm) = pm�1 · ( unit mod p )
�p(p) =p� pp
p
= 1� pp�1
Decreases valuation:
�t =ddt
�t(tn) = n · tn�1
Decreases valuation:
Constants
(K, �)
K� = {c 2 K : �(c) = 0}
R = bZurp
R� = {r 2 R : �p(r) = 0
Constants of a p-derivation:
(ring with der)Constants of a derivation:
(subring)
(submonoid)(ring with p-der)
PART 2
Linear Arithmetic Differential Equations
The poor man’s model companion:
R = bZurp
unramified CDVR, residue field Fp
bZurp = Zp[⇣; ⇣n = 1, p - n]b
unique lift of the Frobenius: �(⇣) = ⇣p
9!� : R ! R
1)
2)
3)
R WILL ALWAYS MEANTHIS RING IN THIS
TALK
Properties:
Simplest Possible Equation:
�x = ↵x
(p)
x = (xij)↵ 2 GLn(R)
x
(p) = (xpij)
Remarks.x 7! �(x)(x(p))�1 almost a cocycle
proof.
↵ 2 gln(R) u0 2 GLn(R)(�u = ↵u(p)
u ⌘ u0 mod p
,
has a unique solution
|x� y|p 1 =) |f(x)� f(y)|p 1p |x� y|p
f : GLn(R) ! GLn(R)
f(x) = �
�1(✏x(p))
Contraction mapping:
Theorem (existence and uniqueness)
�(u) = ✏u(p)�u = ↵u(p) ()
u = limn!1
fn(u0)
✏ = 1 + p↵
matrix norms
(�u = ↵u(p)
u ⌘ u0 mod p
Theorem (coeffs in CDVR)
Ou0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O)
Complete Discrete Valuation Subring=
Proof. O = R�⌫
�⌫(✏) = ✏
�⌫+1(u) = �⌫(✏u(p))
�(�⌫(u)) = ✏(�⌫(u))(p)
�(u) = ✏u(p)
(characterization)
,�⌫(u0) = u0 �⌫(↵) = ↵,
✏ = 1 + p↵
(�u = ↵u(p)
u ⌘ u0 mod p
Theorem (coeffs in valuation delta subring)
O valuation delta-subring =
Proof.
u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)
O0/O finite extension of delta-subrings
Strategy: reformulate as dynamics problem!!
u0,↵, ✏ = 1 + p↵ 2 gln( bO)
9⌫ � 0,�⌫(u) = u=)CDVR regularity Lemma
complete
Theorem (coeffs in valuation delta subring)u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)
O0/O finite extension of delta-subrings
u = �⌫(u)
= �⌫�1(✏u(p))
= · · ·= �⌫�1(✏)(�⌫�2(✏)(· · · (✏u(p)) · · · )(p))(p)
w 2 GLn(R)
(w(p))(m) = (w(m))(p)
(vw)(m) = (vw(m))
w(m) = pick out mth column
1)2)
properties
Strategy: reformulate as dynamics problem!!
Theorem (coeffs in valuation delta subring)u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)
u = �⌫(u)
= �⌫�1(✏u(p))
= · · ·= �⌫�1(✏)(�⌫�2(✏)(· · · (✏u(p)) · · · )(p))(p)
Strategy: reformulate as dynamics problem!!
(w(p))(m) = (w(m))(p)
(vw)(m) = (vw(m))
properties
u(m) = ✏⌫�1(✏⌫�2(· · · (✏u(p)(m)) · · · )
(p))(p)
'(⌘) = ✏⌫�1(✏⌫�2(· · · (✏⌘(p)) · · · )(p))(p)' : An
F ! AnF
u(m) ' : An(Ka) ! An(Ka)fixed point of
Theorem (coeffs in valuation delta subring)u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)
u(m) = ✏⌫�1(✏⌫�2(· · · (✏u(p)(m)) · · · )
(p))(p)
'(⌘) = ✏⌫�1(✏⌫�2(· · · (✏⌘(p)) · · · )(p))(p)' : An
F ! AnFDynamics!
' : An(Ka) ! An(Ka)
# fixed points of < 1
Lemma.
F = Frac(O)
' : An(Ka) ! An(Ka)
# fixed points of < 1
Lemma.
general principle:
' : An(Ka) ! An(Ka)fixed points of ⇢ An(F a)
'/F =)
=) columns are in finite algebraic extensions
Theorem (coeffs in valuation delta subring)u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)
' : An(Ka) ! An(Ka)
# fixed points of < 1
Lemma. (Fornaes and Sibonay)
proof. '(x1, . . . , xn) = (x1, . . . , xn)
Y : 'j(x1, . . . , xn)� x
d�10 xj = 0
homogenize⌘ 7! ✏j⌘
⌘ 7! ⌘(p)built from'
zero dimensional in Y=) Pn
=)' : An(Ka) ! An(Ka)
fixed points of
Y \ {x0 = 0} = ;Claim
={ (1,u) }
Galois TheoryO ⇢ R
Gu/O
{c 2 GLn(O) : 9� 2 AutO(O[u]),� � � = � � �,�(u) = uc}
Galois Group=
, ↵ 2 gln(O)
�u = ↵u(p)
O[u] = O[uij ] = Picard-Vessiot ring= Ring obtained by adjoining entries of u
Gu/O
{c 2 GLn(O) : 9� 2 AutO(O[u]),� � � = � � �,�(u) = uc}
Galois Group of=
�u = ↵u(p)
Alternative Descriptions:
�u/O = {� 2 AutO(O[u]);� � � = � � � 2 GLn(O)}1)
2) 0 ! Iu/O ! O[x, 1/ det(x)] ! O[u] ! 0
StabGLn(R)(Iu/O) ⇠= Gu/O
�u/O ⇠= Gu/O
�u = ↵u(p)
�(uc) = ↵(uc)(p)
Question: For what do we get c 2 GLn(R)
?
�(uc) = u(p)�(c) + �(u)c(p) + p�(u)�(c) + {u, c}⇤
{u, c}⇤ = u(p)c(p)�(uc)(p)
p
Claim.{u, c}⇤ = 0
�c = 01)2)
=) �(uc) = ↵(uc)(p)
W
T
N = WT = TW
G�
= maximal torus of diagonals= permutation matrices
Main Example:
= matrices with roots of unity entries
= permutation matrices with roots ofunity entries
N � = T �W = WT �
GLn(Fa1)
subgroup of
�(uc) = u(p)�(c) + �(u)c(p) + p�(u)�(c) + {u, c}⇤ {u, c}⇤ = u(p)c(p)�(uc)(p)
p
O ⇢ R
O ⇢ R�⌫
Gu/O
delta subring
�u/O = {� 2 AutO(O[u]);� � � = � � � 2 GLn(O)}=
=) Gu/O finite
Theorem.
finite to begin with
u 2 O
Theorem A.9⌦ ⇢ Q2
8↵ 2 Z2 \ ⌦
Gu/O
�u = ↵u(p)
W= finite group containing
“thin set”9u 2 GLn(R),
⌦ ⇢ Zn2
X = {u 2 GLn(R);u ⌘ 1 mod p}
9⌦ ⇢ X
8u 2 X \ ⌦
�(u)(u(p))�1 2 gln(O)
O ⇢ R �-closed subring
Gu/O = N �
R� ⇢ O
Theorem B.
= ball around identity
of the second category
,1)2)
8
=)
9⌦ ⇢ GLn(Ka)
8u 2 GLn(R) \ ⌦u0 = ↵u(p)
dim((Z ·Gu/O)Zar) n
O 3 ↵ �-closed subring of R
Theorem C.Zariski closed
PART 4
Remarks on Picard-Fuchs and Painleve 6 and the Frobenius
E/K
K =
: E(K) ! Ga(K)
ker( ) = E(K)tors
Manin Map:
elliptic curvefunction field
(diff alg map)
E
A1
Et(C) =
Setup:
universal family
E : y2 = x(x� 1)(x� t)
px(x� 1)(x� t)
! = dx/y
R�1
! = ⌘1R�2
! = ⌘2
⌘1(t), ⌘2(t)
H1(Et(C),Z) = Span{�1, �2}
periods
(X,Y ) = ⇤R (X,Y )
dx/y
⇤ = 4t(1� t) d2
dt2 + 4(1� 2t) ddt � 1
Picard-Fuchs Equations:
kills periods
Manin-Map:
Buium’s Picard-Fuchs
⇤ = �2 + ↵1�+ p↵0
(P ) = ⇤
⇣1pr log!(P )
⌘Buium’s Manin Map